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%%文档的题目、作者与日期
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{实变函数练习2.1-2.3}
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\date{2024 年 3 月 18 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设 $E\subseteq \mathbb{R}^n$ 是一个点集，设 $P_0\in E$ 是一个定点。证明下面的三个陈述是等价的：
\begin{enumerate}[label={(\arabic*)}]
\item  $P_0$ 是 $E$ 的聚点。 
\item  在 $P_0$ 的任意邻域内，存在属于 $E$ 但不同于 $P_0$ 的点。 
\item  存在 $E$ 中互异的点 $P_n$, 使得 $P_n\to P_0$. 
\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 02
设 $E\subseteq \mathbb{R}^n$ 是一个点集，$E$ 的全体聚点组成的集合称为 $E$ 的导集，记为 $E'$. 
\begin{enumerate}[label={(\arabic*)}]
\item  设 $A\subseteq B$ 是两个点集，证明 $A'\subseteq B'$. 
\item  对一般的两个点集 $A$ 与 $B$, 证明 $(A\cup B)' = A'\cup B'$. 
\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 03
设 $E\subseteq \mathbb{R}^n$ 是一个点集，$E$ 的全体内点组成的集合称为 $E$ 的开核，记为 $\mathring{E}$. 
设 $A\subseteq B$, 证明 $\mathring{A}\subseteq \mathring{B}$. 

\vspace{0.2cm}

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\item  %Problem 04
设 $E\subseteq \mathbb{R}^n$ 是一个点集，$E$ 和 $E$ 的导集的并集称为 $E$ 的闭包，记为 $\overline{E}$. 
设 $A\subseteq B$, 证明 $\overline{A}\subseteq\overline{B}$. 

\vspace{0.2cm}

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\item  %Problem 05
设 $E\subseteq \mathbb{R}^n$ 不是空集也不是全集，则 $E$ 必有边界点。

\vspace{0.2cm}

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\item  %Problem 06
设 $E=[0,1]\cap \mathbb{Q}$, 求 $E$ 在 $\mathbb{R}$ 中的导集、开核与闭包。

\vspace{0.2cm}

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\item  %Problem 07
设 $E=\{(x,y)\in\mathbb{R}^2 : y=\sin\frac{1}{x},x\in\mathbb{R},x\neq 0\} \cup \{(0,0) \}$, 
求 $E$ 在 $\mathbb{R}^2$ 中的导集 $E'$, 开核 $\mathring{E}$ 与闭包 $\overline{E}$. 

\vspace{0.2cm}

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\item  %Problem 08
设 $E\subseteq \mathbb{R}^n$, 证明开核 $\mathring{E}$ 是开集，导集 $E'$ 与闭包 $\overline{E}$ 是闭集。

\vspace{0.2cm}


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\item  %Problem 9
设 $E$ 是开集，则 $\mathbb{R}^n-E$ 是闭集。设 $E$ 是闭集，则 $\mathbb{R}^n-E$ 是开集。

\vspace{0.2cm}


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\item  %Problem 10
设 $F_1,F_2$ 是 $\mathbb{R}$ 中的两个互不相交的闭集。证明存在两个互不相交的开集 $G_1,G_2$ 可以分别覆盖这两个闭集，即 $F_1\subseteq G_1, F_2\subseteq G_2$. 举例说明存在两个闭集互不相交，但是距离为零。

\vspace{0.2cm}

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\end{enumerate}


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\end{document}

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